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Logarithms are limits

The natural logarithm is the limit of a quotient

 

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Explanation

We start with

Multiplication by h gives

Shifting delivers

Exponentiation gives

Simplification gives

Let , and because h→0 gives that n→∞, and so

The development of this binomium gives

It was all about limits, so we write

On the various terms we apply l'Hôpital's rule several times

Here we see the power series for the exponential function, so

a = e^x    ⇒    x = ln a

so that

 

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